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A singularly altered streamline topology allows faster transport from deformed drops

Published online by Cambridge University Press:  14 October 2024

Pavan Kumar Singeetham Open the ORCID record for Pavan Kumar Singeetham [Opens in a new window] ,
Sumesh P. Thampi Open the ORCID record for Sumesh P. Thampi [Opens in a new window]  and
Ganesh Subramanian Open the ORCID record for Ganesh Subramanian [Opens in a new window]
Pavan Kumar Singeetham
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India
Sumesh P. Thampi
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600036, India
Ganesh Subramanian*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India
*
†Email address for correspondence: [email protected]
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Abstract

We analyse the effect of drop-deformation-induced change in streamline topology on the scalar transport rate (the Nusselt number $Nu$) in an ambient planar linear flow. The drop-phase resistance is assumed dominant, and the drop deformation is characterised by the capillary number ($Ca$). For a spherical drop ($Ca = 0$) in an ambient planar extension, closed streamlines lead to $Nu$ increasing with the Péclet number ($Pe$), from $Nu_0$, corresponding to purely diffusive transport, to $4.1Nu_0$, corresponding to a large-$Pe$ diffusion-limited plateau. For non-zero $Ca$, we show that the flow field consists of spiralling streamlines densely wound around nested tori foliating the deformed drop interior. Now $Nu$ increases beyond the aforementioned primary plateau, saturating in a secondary one that approaches $22.3Nu_0$ for $Ca \rightarrow 0$, $Pe\,Ca \rightarrow \infty$. The enhancement appears independent of the drop-to-medium viscosity ratio. We further show that this singular dependence, of the transport rate on drop deformation, is generic across planar linear flows; chaotically wandering streamlines in some of these cases may even lead to a tertiary enhancement regime.

JFM classification

Mass Transport: Coupled diffusion and flow Drops and Bubbles: Drops Low-Reynolds-number flows: Boundary integral methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 997 , 25 October 2024 , A39
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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References

Acrivos, A. 1971 Heat transfer at high Péclet number from a small sphere freely rotating in a simple shear field. J. Fluid Mech. 46 (2), 233–240.CrossRefGoogle Scholar
Acrivos, A. & Goddard, J.D. 1965 Asymptotic expansions for laminar forced-convection heat and mass transfer. J. Fluid Mech. 23 (2), 273–291.CrossRefGoogle Scholar
Acrivos, A. & Lo, T.S. 1978 Deformation and breakup of a single slender drop in an extensional flow. J. Fluid Mech. 86 (4), 641–672.CrossRefGoogle Scholar
Bajer, K. & Moffatt, H.K. 1990 On a class of steady confined stokes flows with chaotic streamlines. J. Fluid Mech. 212, 337–363.CrossRefGoogle Scholar
Bajer, K. & Moffatt, H.K. 1992 Chaos associated with fluid inertia. In Topological Aspects of the Dynamics of Fluids and Plasmas, pp. 517–534. Springer.CrossRefGoogle Scholar
Banerjee, M.R. & Subramanian, G. 2021 An anisotropic particle in a simple shear flow: an instance of chaotic scattering. J. Fluid Mech. 913, A2.Google Scholar
Barthes-Biesel, D. & Acrivos, A. 1973 Deformation and burst of a liquid droplet freely suspended in a linear shear field. J. Fluid Mech. 61 (1), 1–22.CrossRefGoogle Scholar
Beard, K.V. & Pruppacher, H.R. 1971 A wind tunnel investigation of the rate of evaporation of small water drops falling at terminal velocity in air. J. Atmos. Sci. 28 (8), 1455–1464.2.0.CO;2>CrossRefGoogle Scholar
Bentley, B.J. & Leal, L.G. 1986 a A computer-controlled four-roll mill for investigations of particle and drop dynamics in two-dimensional linear shear flows. J. Fluid Mech. 167, 219–240.CrossRefGoogle Scholar
Bentley, B.J. & Leal, L.G. 1986 b An experimental investigation of drop deformation and breakup in steady, two-dimensional linear flows. J. Fluid Mech. 167, 241–283.CrossRefGoogle Scholar
Brenner, H. 1967 On the invariance of the heat-transfer coefficient to flow reversal in stokes and potential streaming flows past particles of arbitrary shape. J. Math. Phys. Sci. 1, 173–179.Google Scholar
Brignell, A.S. 1975 Solute extraction from an internally circulating spherical liquid drop. Intl J. Heat Mass Transfer 18 (1), 61–68.CrossRefGoogle Scholar
Brooks, B. 2010 Suspension polymerization processes. Chem. Engng Technol. 33 (11), 1737–1744.CrossRefGoogle Scholar
Carslaw, H.S. & Jaeger, J.C. 1948 Conduction of Heat in Solids, p. 200. Clarendon.Google Scholar
Christov, C.I. & Homsy, G.M. 2009 Enhancement of transport from drops by steady and modulated electric fields. Phys. Fluids 21, 083102.CrossRefGoogle Scholar
Cox, R.G. 1969 The deformation of a drop in a general time-dependent fluid flow. J. Fluid Mech. 37 (3), 601–623.CrossRefGoogle Scholar
Cox, R.G., Zia, I.Y.Z. & Mason, S.G. 1968 Particle motions in sheared suspensions XXV. Streamlines around cylinders and spheres. J. Colloid Interface Sci. 27 (1), 7–18.CrossRefGoogle Scholar
Duguid, H.A. & Stampfer, J.F. Jr. 1971 The evaporation rates of small, freely falling water drops. J. Atmos. Sci. 28 (7), 1233–1243.2.0.CO;2>CrossRefGoogle Scholar
Favelukis, M. 2015 Mass transfer around slender drops in an extensional flow: inertial effects at large Péclet numbers. Can. J. Chem. Engng 93 (12), 2274–2285.CrossRefGoogle Scholar
Favelukis, M. 2016 A slender drop in a nonlinear extensional flow. J. Fluid Mech. 808, 337–361.CrossRefGoogle Scholar
Favelukis, M. & Lavrenteva, O.M. 2013 Mass transfer around prolate spheroidal drops in an extensional flow. Can. J. Chem. Engng 91 (6), 1190–1199.CrossRefGoogle Scholar
Favelukis, M. & Lavrenteva, O.M. 2014 Mass transfer around oblate spheroidal drops in a biaxial stretching motion. Can. J. Chem. Engng 92 (5), 964–972.CrossRefGoogle Scholar
Frankel, N.A. & Acrivos, A. 1968 Heat and mass transfer from small spheres and cylinders freely suspended in shear flow. Phys. Fluids 11 (9), 1913–1918.CrossRefGoogle Scholar
Gardiner, C.W. 2004 Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer.CrossRefGoogle Scholar
Goldstein, R.E., Tuval, I. & van de Meent, J. 2008 Microfluidics of cytoplasmic streaming and its implications for intracellular transport. Proc. Natl Acad. Sci. 105 (10), 3663–3667.CrossRefGoogle ScholarPubMed
Guasto, J.S., Rusconi, R. & Stocker, R. 2012 Fluid mechanics of planktonic microorganisms. Annu. Rev. Fluid Mech. 44, 373–400.CrossRefGoogle Scholar
Gupalo, I.P. & Riazantsev, I.S. 1972 Diffusion on a particle in the shear flow of a viscous fluid. Approximation of the diffusion boundary layer. J. Appl. Math. Mech. 36 (3), 447–451.CrossRefGoogle Scholar
Gupalo, I.P., Riazantsev, I.S. & Ulin, V. 1975 Diffusion on a particle in a homogeneous translational-shear flow. Prikl. Mat. Mekh. 39, 497–504.Google Scholar
Hakimi, F.S. & Schowalter, W.R. 1980 The effects of shear and vorticity on deformation of a drop. J. Fluid Mech. 98, 635–645.CrossRefGoogle Scholar
Hinch, E.J. & Acrivos, A. 1980 Long slender drops in a simple shear flow. J. Fluid Mech. 98 (2), 305–328.CrossRefGoogle Scholar
Johns, L.E. Jr. & Beckmann, R.B. 1966 Mechanism of dispersed-phase mass transfer in viscous, single-drop extraction systems. AIChE J. 12 (1), 10–16.CrossRefGoogle Scholar
Juncu, G. 2010 A numerical study of the unsteady heat/mass transfer inside a circulating sphere. Intl J. Heat Mass Transfer 53 (15–16), 3006–3012.CrossRefGoogle Scholar
Kennedy, M.R., Pozrikidis, C. & Skalak, R. 1994 Motion and deformation of liquid drops, and the rheology of dilute emulsions in simple shear flow. Comput. Fluids 23 (2), 251–278.CrossRefGoogle Scholar
Kinzer, G.D. & Gunn, R. 1951 The evaporation, temperature and thermal relaxation-time of freely falling waterdrops. J. Atmos. Sci. 8 (2), 71–83.Google Scholar
Komrakova, A.E., Shardt, O., Eskin, D. & Derksen, J.J. 2014 Lattice Boltzmann simulations of drop deformation and breakup in shear flow. Intl J. Multiphase Flow 59, 24–43.CrossRefGoogle Scholar
Krishnamurthy, D. & Subramanian, G. 2018 a Heat or mass transport from drops in shearing flows. Part 1. The open streamline regime. J. Fluid Mech. 850, 439–483.CrossRefGoogle Scholar
Krishnamurthy, D. & Subramanian, G. 2018 b Heat or mass transport from drops in shearing flows. Part 2. Inertial effects on transport. J. Fluid Mech. 850, 484–524.CrossRefGoogle Scholar
Kronig, R. & Brink, J.C. 1951 On the theory of extraction from falling droplets. Appl. Sci. Res. 2 (1), 142–154.CrossRefGoogle Scholar
Law, C.K. 1982 Recent advances in droplet vaporization and combustion. Prog. Energy Combust. Sci. 8 (3), 171–201.CrossRefGoogle Scholar
Leal, L.G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.CrossRefGoogle Scholar
Liu, A., Chen, J., Wang, Z., Mao, Z. & Yang, C. 2018 Internal mass and heat transfer between a single deformable droplet and simple extensional creeping flow. Intl J. Heat Mass Transfer 127, 1040–1053.CrossRefGoogle Scholar
Liu, A., Chen, J., Wang, Z., Mao, Z. & Yang, C. 2019 Unsteady conjugate mass and heat transfer from/to a prolate spheroidal droplet in uniaxial extensional creeping flow. Intl J. Heat Mass Transfer 134, 1180–1190.CrossRefGoogle Scholar
Magar, V., Goto, T. & Pedley, T.J. 2003 Nutrient uptake by a self-propelled steady squirmer. Q. J. Mech. Appl. Maths 56 (1), 65–91.CrossRefGoogle Scholar
Newman, A.B. 1931 The drying of porous solids: diffusion and surface emission equations. AIChE Trans. 27, 203–220.Google Scholar
Patel, R.P., Patel, M.P. & Suthar, A.M. 2009 Spray drying technology: an overview. Indian J. Sci. Technol. 2 (10), 44–47.CrossRefGoogle Scholar
Polyanin, A.D. 1984 Three-dimensional diffusive boundary-layer problems. J. Appl. Mech. Tech. Phys. 25 (4), 562–571.CrossRefGoogle Scholar
Powell, R.L. 1983 External and internal streamlines and deformation of drops in linear two-dimensional flows. J. Colloid Interface Sci. 95 (1), 148–162.CrossRefGoogle Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Pozrikidis, C. 2002 A Practical Guide to Boundary Element Methods with the Software Library BEMLIB. CRC.CrossRefGoogle Scholar
Prakash, S. & Sirignano, W.A. 1978 Liquid fuel droplet heating with internal circulation. Intl J. Heat Mass Transfer 21 (7), 885–895.CrossRefGoogle Scholar
Rallison, J.M. 1980 Note on the time-dependent deformation of a viscous drop which is almost spherical. J. Fluid Mech. 98 (3), 625–633.CrossRefGoogle Scholar
Rallison, J.M. 1981 A numerical study of the deformation and burst of a viscous drop in general shear flows. J. Fluid Mech. 109, 465–482.CrossRefGoogle Scholar
Rallison, J.M. 1984 The deformation of small viscous drops and bubbles in shear flows. Annu. Rev. Fluid Mech. 16 (1), 45–66.CrossRefGoogle Scholar
Ramachandran, A. & Leal, L.G. 2012 The effect of interfacial slip on the rheology of a dilute emulsion of drops for small capillary numbers. J. Rheol. 56 (6), 1555–1587.CrossRefGoogle Scholar
Rom-Kedar, V., Leonard, A. & Wiggins, S. 1990 An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347–394.CrossRefGoogle Scholar
Rumscheidt, F.D. & Mason, S.G. 1961 Particle motions in sheared suspensions XII. deformation and burst of fluid drops in shear and hyperbolic flow. J. Colloid Sci. 16 (3), 238–261.CrossRefGoogle Scholar
Sabarish, V.N. 2021 Convective transport from drops in complex shearing flows. Master's thesis, Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR).Google Scholar
Sabarish, V.N. & Subramanian, G. 2023 A viscous drop in canonical planar linear flows – effect of deformation on streamline topology. arXiv preprint arXiv:2406.02823.Google Scholar
Stocker, R. 2012 Marine microbes see a sea of gradients. Science 338 (6107), 628–633.CrossRefGoogle Scholar
Stone, H.A. 1994 Dynamics of drop deformation and breakup in viscous fluids. Annu. Rev. Fluid Mech. 26 (1), 65–102.CrossRefGoogle Scholar
Stone, H.A., Nadim, A. & Strogatz, S.H. 1991 Chaotic streamlines inside drops immersed in steady Stokes flows. J. Fluid Mech. 232, 629–646.CrossRefGoogle Scholar
Subramanian, G. & Koch, D.L. 2006 a Centrifugal forces alter streamline topology and greatly enhance the rate of heat and mass transfer from neutrally buoyant particles to a shear flow. Phys. Rev. Lett. 96 (13), 134503.CrossRefGoogle ScholarPubMed
Subramanian, G. & Koch, D.L. 2006 b Inertial effects on the orientation of nearly spherical particles in simple shear flow. J. Fluid Mech. 557, 257–296.CrossRefGoogle Scholar
Subramanian, G. & Koch, D.L. 2006 c Inertial effects on the transfer of heat or mass from neutrally buoyant spheres in a steady linear velocity field. Phys. Fluids 18 (7), 073302.CrossRefGoogle Scholar
Subramanian, G. & Koch, D.L. 2007 Heat transfer from a neutrally buoyant sphere in a second-order fluid. J. Non-Newtonian Fluid Mech. 144 (1), 49–57.CrossRefGoogle Scholar
Taylor, G.I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. A 146 (858), 501–523.Google Scholar
Torza, S., Henry, C.P., Cox, R.G. & Mason, S.G. 1971 Particle motions in sheared suspensions. XXVI. Streamlines in and around liquid drops. J. Colloid Interface Sci. 35 (4), 529–543.CrossRefGoogle Scholar
Vivaldo-Lima, E., Wood, P.E., Hamielec, A.E. & Penlidis, A. 1997 An updated review on suspension polymerization. Ind. Engng Chem. Res. 36 (4), 939–965.CrossRefGoogle Scholar
Vlahovska, P.M., Bławzdziewicz, J. & Loewenberg, M. 2009 Small-deformation theory for a surfactant-covered drop in linear flows. J. Fluid Mech. 624, 293–337.CrossRefGoogle Scholar
Watada, H., Hamielec, A.E. & Johnson, A.I. 1970 A theoretical study of mass transfer with chemical reaction in drops. Can. J. Chem. Engng 48 (3), 255–261.CrossRefGoogle Scholar
Wegener, M., Paul, N. & Kraume, M. 2014 Fluid dynamics and mass transfer at single droplets in liquid/liquid systems. Intl J. Heat Mass Transfer 71, 475–495.CrossRefGoogle Scholar
Wilkie, J. 2004 Numerical methods for stochastic differential equations. Phys. Rev. E 70 (1), 017701.CrossRefGoogle ScholarPubMed